Pair covering designs with block size 5

In this article we look at pair covering designs with a block size of 5 and v≡0(mod4). The number of blocks in a minimum covering design is known as the covering number C(v,5,2)C(v,5,2). For v⩽24v⩽24, these values are known, and all but v=8v=8 exceed the Schönheim bound, L(v,5,2)=⌈v/5⌈(v-1)/4⌉⌉L(v,5,2)=⌈v/5⌈(v-1)/4⌉⌉. However, for all v⩾28v⩾28 with v≡0(mod4), it seems probable that C(v,5,2)=L(v,5,2)C(v,5,2)=L(v,5,2). We establish this for all but 17 possible exceptional values lying in the range 40⩽v⩽28040⩽v⩽280.